The budget constraint defines what a consumer can afford given prices and income. The consumer's problem is to pick the best affordable bundle, which means finding where the highest indifference curve just touches the budget line. The method for solving this depends on the utility function: tangency (MRS = price ratio) for smooth preferences like Cobb-Douglas, and the kink condition for perfect complements.
Difficulty: Intermediate | Prerequisites: utility functions and indifference curves (the first set of study notes in this series), basic algebra.
You now know how to describe what a consumer wants (utility functions, indifference curves). The next question is: what can they actually get? The budget constraint draws the boundary between affordable and unaffordable bundles. Consumer optimisation combines the two pieces: lay the indifference map over the budget line and find the best bundle the consumer can reach. This is the core analytical exercise of the first half of ECON 302, and the technique you learn here (tangency or kink-matching) appears in nearly every problem set and exam.
Budget constraint
The equation Px·X + Py·Y = M, describing all bundles that exactly exhaust the consumer's income M at prices Px and Py. Think of it as the line between "can afford" and "cannot afford" on a graph.
Budget set
All bundles at or below the budget constraint: every combination (X, Y) with Px·X + Py·Y ≤ M. The feasible region, in other words.
Slope of the budget constraint
Equals -Px/Py. This is the market's exchange rate between the two goods: to get one more unit of X, the consumer must give up Px/Py units of Y.
Intercepts
The maximum quantity of one good the consumer could buy if they spent everything on it. X-intercept = M/Px. Y-intercept = M/Py.
Optimal bundle (consumer's choice)
The bundle that maximises utility subject to the budget constraint. For smooth, convex preferences (Cobb-Douglas), this is where MRS = Px/Py and the budget constraint binds. For perfect complements, it is where the kink line meets the budget constraint.
Tangency condition
At the optimum for smooth preferences, the indifference curve is tangent to the budget line. Formally, MRS = Px/Py. The consumer's willingness to trade matches the market's rate of exchange.
Rewrite Px·X + Py·Y = M in slope-intercept form: Y = (M/Py) - (Px/Py)·X.
The Y-intercept is M/Py (spend everything on Y). The X-intercept is M/Px (spend everything on X). Connect them with a straight line.
The slope is -Px/Py. This is always negative (downward-sloping) because buying more X requires giving up some Y.
Example: M = 250, Px = 10, Py = 5. X-intercept = 250/10 = 25. Y-intercept = 250/5 = 50. Slope = -10/5 = -2. For every additional unit of X, the consumer gives up 2 units of Y.
For U = X^a · Y^b, the standard approach uses two equations:
Tangency condition: MRS = Px/Py. For this utility function, (a/b)(Y/X) = Px/Py.
Budget constraint: Px·X + Py·Y = M.
Solve the tangency condition for Y in terms of X (or vice versa), substitute into the budget constraint, and solve.
There is also a shortcut for Cobb-Douglas. The optimal quantities are:
X* = (a / (a + b)) · (M / Px)
Y* = (b / (a + b)) · (M / Py)
The consumer spends a fraction a/(a+b) of income on X and b/(a+b) on Y. This is a defining property of Cobb-Douglas preferences: the expenditure shares are constant.
Example: U = X · Y (so a = 1, b = 1), M = 200, Px = 2, Py = 4.
X* = (1/2)(200/2) = 50 units.
Y* = (1/2)(200/4) = 25 units.
Check: 2(50) + 4(25) = 100 + 100 = 200. The budget is exhausted.
For U = min(aX, bY), the consumer always chooses the kink point where aX = bY, because any other point on the budget line wastes money on a good that yields no extra utility.
The method:
Set the kink condition: aX = bY, so Y = (a/b)X.
Substitute into the budget constraint: Px·X + Py·(a/b)·X = M.
Solve for X, then find Y.
Example: U = min(2X, Y), M = 100, Px = 1, Py = 2.
Kink: 2X = Y.
Budget: 1·X + 2·(2X) = 100, so 5X = 100, X = 20.
Y = 2(20) = 40.
Check: 1(20) + 2(40) = 20 + 80 = 100.
P_X \cdot X + P_Y \cdot Y = MBudget constraint. Rearranged: Y = M/P_Y - (P_X/P_Y) \cdot X.
\text{Slope} = -\frac{P_X}{P_Y}The slope of the budget line. Negative because the trade-off runs in opposite directions.
X^* = \frac{a}{a+b} \cdot \frac{M}{P_X}, \quad Y^* = \frac{b}{a+b} \cdot \frac{M}{P_Y}Cobb-Douglas shortcut for U = X^a · Y^b. Memorise this; it saves time on exams.
aX = bY \quad \text{(kink condition for } U = \min(aX, bY)\text{)}Perfect-complements optimisation starts here. Substitute into the budget constraint to solve.
Students often report the slope of the budget constraint as a positive number. The slope is -Px/Py, always negative. The question "what is the slope?" expects the sign.
Students sometimes try to use the tangency condition MRS = Px/Py for perfect complements. There is no tangency with L-shaped curves. Use the kink condition instead.
A common algebra error with the Cobb-Douglas shortcut: students mix up which exponent goes with which good. For U = X^a · Y^b, the share of income spent on X uses the exponent a, not b.
Students sometimes forget to verify the budget constraint binds. After solving, always check that Px·X* + Py·Y* = M exactly.
⚠️ Graphing the budget constraint and labelling both intercepts is a standard exam question. Know M/Px and M/Py cold.
⚠️ The slope of the budget constraint (-Px/Py) is asked on almost every problem set. Include the negative sign.
⚠️ Expect at least one optimisation problem per exam. You need to recognise the utility function type (Cobb-Douglas vs. perfect complements) and apply the correct method without prompting.
⚠️ The Cobb-Douglas shortcut (constant expenditure shares) is a major time-saver. Verify your answer satisfies the budget constraint as a final check.
Fill in the blank: if M = 300, Px = 15, and Py = 10, the X-intercept is ____ and the Y-intercept is ____.
True or false: the slope of the budget constraint depends on income.
Fill in the blank: for U = X^2 · Y^3, the fraction of income spent on X is ____.
True or false: for perfect complements U = min(3X, Y), the kink condition is Y = 3X.
Fill in the blank: using the Cobb-Douglas shortcut for U = X · Y with M = 100 and Px = 5, X* = ____.
Answers: 1. X-intercept = 20, Y-intercept = 30. 2. False (slope = -Px/Py, independent of M). 3. 2/5 (since a = 2, b = 3, share = 2/(2+3)). 4. True. 5. X* = (1/2)(100/5) = 10.
Q: A consumer has income M = 400, Px = 8, Py = 10. What are the intercepts and slope of the budget constraint?
A: X-intercept = 400/8 = 50. Y-intercept = 400/10 = 40. Slope = -8/10 = -0.8.
Q: For U = X^2 · Y and M = 600, Px = 4, Py = 6, find the optimal bundle using the Cobb-Douglas shortcut.
A: a = 2, b = 1. X* = (2/3)(600/4) = 100. Y* = (1/3)(600/6) = 33.33. Check: 4(100) + 6(33.33) = 400 + 200 = 600.
Q: For U = min(2X, Y), M = 100, Px = 1, Py = 2, find the optimal bundle.
A: Kink condition: 2X = Y. Budget: X + 2(2X) = 100, so 5X = 100, X* = 20. Y* = 40. Check: 1(20) + 2(40) = 100.
Q: A consumer's utility function is U = X · Y. She has M = 200, Px = 2, Py = 4. How much does she spend on each good?
A: With a = b = 1, she spends half her income on each good. Spending on X = 100, spending on Y = 100. So X* = 100/2 = 50, Y* = 100/4 = 25.
Q: Why can you not use tangency (MRS = Px/Py) for U = min(3X, 2Y)?
A: The indifference curves are L-shaped, so there is no smooth tangent line at the kink. The MRS is undefined at the corner. You must use the kink condition 3X = 2Y instead.
Budget constraints reappear throughout the course whenever a new constraint is introduced: time constraints in labour-leisure models, intertemporal budget constraints in savings/borrowing problems, and isocost lines in producer theory (where the same Px·X + Py·Y = M logic applies to inputs and costs).
The optimisation technique here (tangency for smooth curves, kink for complements) is the same method used in producer theory to find the cost-minimising input bundle, with MRTS replacing MRS and the isocost line replacing the budget line.
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